## Introduction

The Moore’s law now faces the bottleneck due to the short channel effect of Si-based metal-oxide-semiconductor field field-effect transistors (MOSFET)1,2. Considering the prospect of integrated circuits (ICs), two main technology roadmaps have been put forward by the industry field, namely more Moore roadmap and more than Moore roadmap3,4. The more Moore roadmap aims to pursue the device scaling complying with the Moore’s law3, while the more than Moore technology focuses on improving the functionality of ICs, such as combining digital electronics with devices such as radio frequency (RF), power devices and sensors4. In the more Moore field, unlike the thickness-dependent mobility of Si and Ge channels, two dimensional (2D) materials not only have atomically thin channel thickness which is beneficial for the gate control, but also maintain promising carrier mobility5,6,7,8. Previous works have proven that devices using 2D materials are capable to fulfill the International Technology Roadmap for Semiconductors (ITRS) requirements9 even with nanometer-scale gate length10,11,12,13,14. Recently, the transistor channel length has already scaled down to 5 nm node15. How to further decrease the power consumption and sustain the Moore’s law is a task of top priority. Various novel transistors such as Fin filed-effect-transistors (FinFETs)16, fully depleted silicon on insulator (FDSOI) transistors17, tunneling FET (TFETs)18,19 and negative capacitance (NC) transistors20,21 have been put forward. Apart from novel device configurations, looking for new materials is an alternative approach to sustain the Moore’s law. Recently, cold-source FETs (CS-FETs) have been proposed to achieve sub-60 mVdec−1 subthreshold swing (SS), which can be realized by using materials with desired density of state (DOS) such as Dirac materials22, appropriately doped semiconductors23,24 and materials with gaps near the Fermi level (εF)25. Compared with the complex heterostructure of Dirac source materials or appropriately doped semiconductors, the ‘cold’ metal 2H MS2 (M = Nb, Ta) with gaps close to the εF, equivalent to a naturally p-doped or n-doped semiconductor, would be an ideal solution to fulfill the steep slope of SS25. The unique band structure allows the ‘cold’ metal MOSFET (CM-FET) to filter the transmission of high-energy carriers in the subthreshold region and reach sub-60 mVdec−1 SS22. Another cornerstone is that the ‘cold’ metal monolayer 2H NbS2 and TaS2 have been successfully synthesized26,27 and served as the injection source in the heterojunction transistors28, which solids the way for the research of ‘cold’ metal heterojunctions and transistors.

In this work, we conduct a comprehensive electronic and transport calculation of ‘cold’ metals (NbS2 and TaS2) and their devices with transition metal dichalcogenide (TMD) channels. The SS of CM-FETs successfully breakthrough the 60 mVdec−1 thermionic limit at room temperature. In terms of more Moore field (to extend the Moore’s law), CM-FETs are capable to fulfill the on-state current, power consumption and cut-off frequency (fT) requirements of both ITRS high performance (HP) and low dissipation (LP) goals. Apart from the favorable performance against ITRS goals, the CM-FETs with unique band structures of ‘cold’ metals can successfully achieve the negative differential resistance (NDR) effect, which can be further exploited in multifunctional ICs. Owing to the wide transmission path in the broken gap characters of MS2/MoS2 heterojunctions, the peak current is several orders of magnitude higher than the typical tunneling diodes29,30, with the mechanism shown in Supplementary Fig. 1. The benchmarking 4110 μAμm−1 peak current and 1.1×106 peak-valley ratio (PVR) are achieved by NbS2/MoS2 and TaS2/MoS2 CM-FETs, respectively. The results demontrated that the superior Ion, SS, fT and NDR effect are obtained by our CM-FETs simultaneously, which provides a feasible method for the development of state-of-the-art logic devices beyond the Moore’s law.

## Results and discussion

### The performance of CM-FETs against the ITRS goals

The calculated lattice constants of monolayer 2H NbS2 and TaS2 are a1 = 3.36 Å and a2 = 3.31 Å respectively, which is in agreement with previous results of lattice constants28,31. The band structures and DOS of 2H NbS2 and TaS2 are shown in Fig. 1. Energy gaps occur above (ECG) and below (EVG) the εF of ‘cold’ metals. The NbS2 and TaS2 can thus be considered as heavily p-doped or n-doped semiconductors and used as the injection source of a transistor. Considering the DOS between the εF and EVG is higher than the DOS above the εF, the ‘cold’ metal is more suitable to serve as the source electrode in a p-type transistor. Previous work presented that MX2 CM-FETs with 10 nm gate length have successfully reached the sub-60 mVdec−1 SS at room temperature25. To illustrate the mechanism of superior SS, schematic energy band diagram comparing with the p-type conventional MOSFET and CM-FET in off-state are shown in Fig. 1(c, d). In terms of the conventional p-type MOSFET in Fig. 1c, the DOS of source electrode varies inversely with the energy. In the off-state, electrons have a long thermal due to the thermal Boltzmann distribution (n(E) = D(E)f(E)), leading to the 60 mVdec−1 limit on SS. Unlike the n(E) of normal source having a long thermal tail, the thermal tail of ‘cold’ metal source below εS is filtered by the D(E) around EVG22. The current of cold metal device is calculated by the Landauer–Bűttiker formula32,

$$I = \frac{{2e}}{h}{\int}_{{{{\mathrm{ - }}}}\infty }^{{{{\mathrm{ + }}}}\infty } {\{ T(E)D(E)[f_{{{\mathrm{S}}}}(E - \varepsilon _{{{\mathrm{S}}}}) - f_{{{\mathrm{D}}}}(E - \varepsilon _{{{\mathrm{D}}}})]\} } {{{\mathrm{d}}}}E$$
(1)

where T(E) is the transmission coefficient, fS and fD are the Fermi-Dirac distribution functions for source and drain electrodes, εS and εD are the Fermi levels of source and drain electrodes, respectively. In formula (1) the current is proportional with the DOS. The n(E) of ‘cold’ metal source, steeper than the f(E), decreases super-exponentially with the decreasing of energy, which allows the CM-FET to achieve the sub-60 mVdec−1 SS. Meanwhile, electrons localized around the εS permit a large on-state current. The schematic energy band diagrams of the CM-FET corresponding to the on-state and the off-state are shown in Supplementary Fig. 2. As the transistor channel length scales down, it is of interest to investigate whether the superior SS can be maintained in the sub-5 nm node CM-FETs.

The schematic view of MS2/TMDs CM-FET and the corresponding I-V curves are shown in Fig. 2. The source and drain electrodes are using ‘cold’ metal and p-doped TMD materials, respectively, and the heterojunction of ‘cold’metal and TMD materials is shown in Supplementary Fig. 3. The p-type doping concentration of source is 3×1013 cm−2. The gate length of CM-FET is 5 nm. The source and drain length is 1 nm. The bias voltage is 0.64 V. The gate oxide is SiO2 and the equivalent oxide thickness (EOT) is 0.41 nm based on the ITRS standards in 20139. Considering the top contact configuration in Fig. 2a is more feasible than the edge contact configuration shown in Supplementary Fig. 4, we only calculate the performance of top contact CM-FETs here. I-V curves and performance benchmark of edge contact CM-FETs are shown in Supplementary Fig. 4 and Supplementary Fig. 5, respectively. On-state current (Ion) is obtained by the corresponding on-state gate voltage (VGS(on)). The VGS(on) is defined as VGS(on) = VGS(off) + Vdd, where VGS(off) is the off-state gate voltage defined by ITRS standards in 20139. Vdd is the bias voltage applied between the source and drain electrodes, fixed at 0.64 V in our work. We observed that SS of CM-FETs significantly outperforms that of heavily p-doped TMD MOSFETs. Among these studied CM-FETs, NbS2 CM-FETs exhibit lower SS because the EVG is close to the εF, while TaS2 CM-FETs deliver larger currents owing to the wide transmission path between the EVG and εF as shown in Fig. 1b. The most favorable SS (45 mVdec−1) and Ion (2643 μAμm−1) achieved by NbS2/MoSe2 and TaS2/ MoS2 CM-FETs, respectively prove the analysis above. The CM-FET with smaller bais Vdd = 0.05 V is shown in Supplementary Fig. 6. Compared with the I-V curves of Vdd = 0.64 V, the current of Vdd = 0.05 V is obviously degraded due to the small bias voltage. The largest Ion of 53 μAμm−1 is achieved by TaS2/ MoS2 CM-FETs. The SS of CM-FET is slightly improved compared with that of Vdd = 0.64 V, with the smallest SS of 45 mVdec−1 achieved by NbS2/MoSe2 CM-FETs. The performance of CM-FET compared with IRDS 2018 requirements is summarized in Supplementary Table 133. The top contact CM-FETs with smaller SS and Ion than that of edge contact CM-FETs (data shown in Supplementary Fig. 4 and Supplementary Fig. 5) indicates that barriers of top contact devices interacted by the van der Waals force is higher than that of edge contact configurations. So, with the extreme low SS, the CMFET proposed in this work can fully sustain the more Moore roadmap in the 5 nm nodes.

To present a panoramic performance analysis of CM-FETs in 5 nm node, Ion and SS of our work compared with previous work are shown in Fig. 334,35,36,37,38,39,40,41. The Ion of CM-FETs ranges from 693 μAμm−1 to 2643 μAμm−1 and TaS2/MoS2 CM-FET with largest Ion is about three times higher than that of ITRS HP goals. The Ion performance is only inferior to the monolayer arsenene and antimonene MOSFETs (Ion of 3200 μAμm−1 and 2980 μAμm−1, respectively) and comparable with the black phosphorus (BP) MOSFET with 1994 μAμm−1 Ion. As is known, the BP is limited to the poor air-stability42. The superior Ion of arsenene and antimonene MOSFETs benefits from the ideal heavily doped source/drain electrodes34. As for SS, the performance of CM-FETs is overall superior to previous results in Fig. 3. Especially, the NbS2/MoSe2 and TaS2/MoSe2 CM-FETs even achieve the low SS of 45 and 47 mVdec−1, respectively at room temperature. In Fig. 3c we plot the power dissipation (PDP) and cut-off frequency (fT). PDP is a decisive parameter for low dissipation applications, defined as $${{{\mathrm{PDP}}}} = V_{{{{\mathrm{DS}}}}}(Q_{{{{\mathrm{on}}}}} - Q_{{{{\mathrm{off}}}}})/w$$, where Qon and Qoff are the total charge of the channel in on-state and off-state, respectively. w is the channel width. The PDP values of CM-FETs vary from 0.144 to 0.197 fJum−1, obviously lower than the ITRS requirement of 0.24 fJum−1, showing a desired gate control capability. fT is a relevant factor for radio frequency devices, obtained by $$f_{{{\mathrm{T}}}} = g_{{{\mathrm{m}}}}/(2\pi C_{{{{\mathrm{total}}}}})$$, where gm is the transconductance of MOSFETs, defined as $$g_{{{\mathrm{m}}}} = I_{{{{\mathrm{DS}}}}}/V_{{{{\mathrm{GS}}}}}$$. Ctotal is the sum of gate capacitance CG and fringing capacitance (Cf). Based on the ITRS standard, the gate-to-source/drain overlapped parasitic capacitance and Miller effect are all included in the Cf that is assumed as twice of the CG. CG is the gate capacitance, defined as $$C_{{{\mathrm{G}}}} = \partial Q_{{{{\mathrm{ch}}}}}/\partial V_{{{{\mathrm{GS}}}}}$$, where Qch is the charge in gate electrode region. fT of NbS2/MoS2 and TaS2/MoS2 in Fig. 3c reach the THz level, indicating that our CM-FETs are competent to be applied into radio frequency circuits43. Hence, the air-stability and excellent performance totally strengthen the competitive advantage of CM-FETs among these low-dimensional material MOSFETs as listed in Fig. 3.

To better understand the mechanism of extreme low (sub-60 mVdec−1) SS achieved by the CM-FETs, we take the NbS2/MoSe2 CM-FET as an example to plot the projected local density of states (PLDOS) in Fig. 4. The SS ranges from 45 mVdec−1 to 59 mVdec−1 with VGS increasing from −0.2 V to 0 V in Fig. 4a. The PLDOS in Fig. 4b and c represents the switching process of the device. At VGS = −0.2 V, the ΦB is small. Thermal emission current can directly transport through the channel region. With the increasing of VGS, ΦB and EVG are overlapped. Therefore, the current is only tunneling around the top of EVG, so the transmission efficiency, T(E), around the εS decreases rapidly compared with its counterpart in Fig. 4b.

### The NDR effect of CM-FETs

The device structure and simulation methods for NDR effect is the same as the 2.1 section. Apart from the superior Ion and SS, CM-FETs are demonstrate the NDR effect, Based on the PLDOS in Fig. 4, the ‘cold’ metal source can form the type-III band alignment with the heavily p-doped MoS2. I-V curves of NbS2/MoS2 and TaS2/MoS2 CM-FETs under various VGS are shown in Fig. 5. Results claim that the NbS2/MoS2 CM-FET tends to deliver a large peak current from 1791 μAμm−1 to 4110 μAμm−1, while the large peak-valley ratio (PVR) is ready to be obtained by the TaS2/MoS2 CM-FET with the detailed values labelled in Fig. 5. The NbS2/MoS2 CM-FET has the largest peak current 4110 μAμm−1 at VGS = −2V. The largest PVR of 1.1×106 is achieved by the TaS2/MoS2 CM-FET at VGS = −1V, several orders of magnitude higher than the mainstream reports, which will be discussed later. It is noteworthy that the VGS plays an important role in controlling the peak current and peak-valley ratio (PVR). The peak current varies inversely with the VGS, while the PVR is proportional with the VGS.

We plot the energy band diagrams at VGS = −1V in Fig. 5c to analyze the NDR mechanism in the NbS2/MoS2 CM-FET. The I-IV in Fig. 5c represent the energy band diagrams under various bias voltages (VDS). Notably, unlike the conventional type-III band alignment NDR device with a narrow transmission path between the broken gap44,45, the wide transmission path, as shown in Supplementary Fig. 7, allows the CM-FET to achieve a larger current, which is desirable for the extremely high PVR. Considering the εS is already tuned to the valence band maximum (VBM) of MoS2 with 0.2 V VDS, the current reaches the peak point. With the increasing of VDS, the current comes to the valley point, because the transmission is blocked by the overlapped gaps of NbS2 and MoS2. As the VDS further increases, the current starts to rise with a transmission path appearing below the VBM of MoS2. It can be concluded that the width of energy window between the εD and VBM of MoS2 is a key factor for the peak current. Considering lower VGS corresponds to a wider transmission path as shown in Supplementary Fig. 7, the peak current is inversely proportional with the VGS. In terms of the PVR, with the decreasing of VGS, a larger VDS is needed to shift the gap of MoS2 and reach valley point. The εD in valley point is even close to the VBM of NbS2, which leads to more efficient transmission through the VBM of MoS2 and a larger valley current. Hence, the PVR is proportional with the VGS.

Furthermore, to analyze the relationship between temperature and NDR effect, I-V curves of NbS2/MoS2 CM-FET with VGS = −1V at various temperature are shown in Fig. 6. The corresponding analysis of TaS2/MoS2 CM-FET is shown in Supplementary Fig. 8. Encouragingly, the NDR effect is improved with the decreasing of temperature. As temperature decreases in Fig. 6b, the peak current increases, while the valley current decreases. Consequently, the PVR(peak current) increases from 549(989 μAμm−1) to 2.3×106(1070 μAμm−1) with temperature decreasing from 300 K to 100 K. Based on Eq. (2), the current is mainly contributed by transmission between the εS and εD of device. As temperature decreases in Fig. 6f, the Fermi-Dirac distribution near the εF becomes sharp. Thus, the weight of current contributed by the transmission between εS and εD is even higher, while the weight of transmission below the εS and above the εD becomes lower. In terms of peak current in Fig. 6d, there is large T(E) around the εS. On the contrary, the T(E) of valley current between εS and εD is really flat. As a result, the valley current is proportional with the temperature and peak current varies inversely with the temperature. These results claim that the NDR performance of NbS2/MoS2 CM-FET is immune to the temperature oscillation, which enables the NbS2/MoS2 CM-FET adapt to more complex working environment.

Apart from the temperature effect, PVR and peak current are regarded as two decisive factors for practical applications. The NDR performance of NbS2/MoS2 and TaS2/MoS2 CM-FETs compared with previous results is summarized in Fig. 744,45,46,47,48,49,50,51,52. To deliver a fair comparison, the device width is normalized to 10μm for both our devices and data from previous reports. The PVR of NbS2/MoS2 CM-FET is only inferior to the MoS2/WSe2 and BP/Al2O3/BP heterojunction NDR devices. However, the poor air-stability of BP and the small peak current of MoS2/WSe2 NDR device hinder their practical application. Compared with previous results superior in either PVR or peak current, the NDR devices in our work are fully capable to achieve both large PVR and peak current simultaneously. Especially, the peak currents in this work are several orders of magnitude higher than previous work in Fig. 7. The large peak current not only improves the noise margin ability of NDR devices and relevant circuits, but also increases the output power and enhances the stability of oscillation circuits. With the large peak current and PVR, NbS2/MoS2 and TaS2/MoS2 CM-FETs are suitable for the MVL system and multifunctional device in more than Moore field.

In summary, we present a comprehensive electronic and transport calculation of nanoscale CM-FETs based on the ‘cold’ metal NbS2 and TaS2 heterojunctions. The Ion of CM-FETs with 5 nm channel length can achieve ITRS HP and LP goals simultaneously to further sustain the more Moore roadmap, which are rarely fulfilled in previous work because of the contradictory requirements of HP (high drain current) and LP (small SS). The dilemma is successfully solved by the superior switching performance of ‘cold’ metals and the favorable mobility of 2D TMDs. The extreme low SS of 45 and 47 mVdec−1 is obtained for NbS2/MoSe2 and TaS2/MoSe2 CM-FETs, respectively In terms of NDR effect in more than Moore field, our results claim that the large peak current and PVR can be both achieved by NbS2/MoS2 and TaS2/MoS2 CM-FETs, owing to the wide broken gap feature. We find that the peak currents and PVR can be effectively controlled by the gate voltage and immune to the temperature influence. The recording 4110 μAμm−1 peak current and 1.1×106 PVR are achieved by NbS2/MoS2 and TaS2/MoS2 CM-FETs, respectively. The results prove that ‘cold’ metal materials are competitive candidates for multifunctional logic devices and can be employed into the MVL system and radiofrequency circuits.

## Methods

### Calculation method

We calculated the electronic and transport properties of ‘cold’ metal MS2 and their devices with TMD channels based on density functional theory (DFT) and non-equilibrium Green function (NEGF) with Atomsitix Tool Kit 2020 package53. The exchange correlation was the Perdew-Burke-Ernzerhof (PBE) functional of generalized gradient approximation (GGA). All simulations are conducted with the Pseudo Dojo pseudopotential and DFT-D3 van der Waals correction54. A 80-Hartree cut-off energy was adopted. Monkhorst-Pack grids used for the transport calculation sampling the 8×1×163 k point meshes. To avoid the interaction from adjacent layers, a 30 Å vacuum was applied to the device for transport calculations. The other basic settings refer to our previous work55,56,57. In ATK platform, the transmission is calculated by the non-equilibrium Green’s function (NEGF) method. The key quantity to calculate is the retarded Green’s function matrix G(E), which can be described as58

$$G(E) = [ES - H - \sum _S(E) - \sum _D(E)]^{ - 1}$$
(2)

where E is the energy. S and H are the overlap and Hamiltonian matrices, respectively. $$\sum ^S(E)$$ and $$\sum ^D(E)$$ are self-energy of source and drain electrodes, respectively. The transmission coeffcient T(E) at given energy E is represented by59

$$T(E) = trace{{{\mathrm{[}}}}\Gamma _S{{{\mathrm{(}}}}E{{{\mathrm{)}}}}G{{{\mathrm{(}}}}E{{{\mathrm{)}}}}\Gamma _D{{{\mathrm{(}}}}E{{{\mathrm{)}}}}G^{\dagger} {{{\mathrm{(}}}}E{{{\mathrm{)]}}}}$$
(3)

where G(E) and $$G^{\dagger} (E)$$ are the retard and advanced Green’s functions, respectively. $$\Gamma _{S/D}(E)$$ is the level broadening source/drain electrodes, defined as $$\Gamma _{S/D}(E) = i[\mathop{\sum}\limits_{S/D}(E) - \mathop{\sum}\limits_{S/D}(E)]$$.

The carrier density can be included into the Poisson equation to renew the electrostatic potential through60

$$\nabla ^2\varphi (r) = \frac{e}{{4\pi \varepsilon _0}}n(r)$$
(4)

where φ(r) is the electrostatic potential. e is the unit charge. ε0 is the vacuum permittivity. n(r) is the valence electron density. Then the self-consistent iteration procedure is constructed between the Schrodinger and Poisson equations.

### Accuracy verification

To confirm the accuracy of our simulation, We compared the simulated single gate monolayer MoS2 MOSFET with the experimental results in Supplementary Fig. 961. The detailed parameters of experimental and simulated monolayer MoS2 MOSFET are shown in Supplementary Table 2 and Supplementary Table 3, respectively. The simulated SS of 209 mVdec−1 is in good agreement with the experimental one of 208 mVdec−1 in the MoS2 MOSFET. Such an agreement validates the reliability of our simulations. The Poisson–Schrodinger solver is shown in Supplementary Fig. 10. To illustrate the SS improved by the 2D material MOSFETs compared with the Si-based MOSFETs, the schematic view of Si-based MOSFET, simulated I-V curves and PLDOS compared with monolayer MoS2 MOSFET are shown in Supplementary Fig. 11, Supplementary Fig. 12 and Supplementary Fig. 13, respectively. The simulated parameters are summarized in Supplementary Table 4.